Abstract : In recent years, considerable progress has been made in exact linear
algebra which offered us a significant number of different algorithms
for each problem. This dissertation explores adaptive algorithms as a
means to get effective solutions. As a result, we obtain very fast
solutions in practice, the average complexity of which is comparable
to the state of the art.
In this thesis we consider the modeling and the use of adaptive
algorithms on the example of the computation of the integer
determinant, of rational algorithms and of the Smith normal form.
As a measure of the performance of adaptive algorithms, we propose the
expected complexity and validate our conclusions with an
experimental evaluation.
To build an adaptive algorithm, we use discovered characteristics of
matrices and the explicit comparison of partial computation times. Our
algorithms are based on the Chinese Remaindering algorithm with full
and rational reconstruction and preconditioning. Our goal is to exploit
the early termination condition faster than other algorithms.
To improve the algorithms, we can also use heuristics. In this thesis
we propose a heuristic scheme for calculating the Smith normal form of
an exact sequence of matrices. Instead of treating each matrix
separately, we show that the elimination of a row or column of matrix
causes reductions in the neighboring matrices. Thus, we succeeded in
computing some Smith forms for whole sequences, despite the fact that
it was not possible for any of the larger matrices taken separately.